Lecture 8 – Integration Basics Functions – know their shapes and properties A few (very few) examples:

Trigonometric Rules Know basics about sine, cosine, tangent, secant, plus 2 right triangles Beyond these angles: and use reference angles for all quadrants.

Substitution Rule First approach for any integral should be a u-substitution. Ex. 1 Which (if any) of the following can use a basic u-substitution?

Ex. 2 Which (if any) of the following can use a basic u-substitution?

Know derivatives for trig functions. Ex Know derivatives for trig functions. Ex. 3 But what about antiderivatives?

Ex.4 What antiderivative for secant function?

Ex.5

Need to try a different 1.

Lecture 9 – Integration By Parts U-substitution is the reverse of the chain rule. Likewise, by parts is the “almost” reverse of the product rule.

When figuring out integrals, now looking for one of the following: 1: know the 2: look for 3: look for 4: look for When trying to decide what to use for the u,

Example 1

Example 2

Example 3

Example 4 What is needed to solve each?

Lecture 10 – More Integration By Parts Example 5

Example 6 What is needed to solve each?

Example 7

Example 8

Lecture 11 – Trig Integrals Use u-sub, trig identities, and/or by parts.

Trig identities:

Example 1

Example 2

Example 3 With dealing with sine or cosine functions, you are looking for (cos x dx) or (sin x dx), respectively.

Example 4

Lecture 12 – More Trig Integration With dealing with tangent or secant, you are looking for (sec2 x dx) or (sec x tan x dx), respectively. Example 5

Example 6

Example 7

Trig Substitution When faced with one of the above in an integral , create a right triangle and substitute trig expressions in  for algebraic expressions of x. (unless a simple u-substitution is available)

Lecture 13 –Trig Substitution Example 1

Example 2

Example 3

Lecture 14 – Partial Fractions Combine the following:

Process can be reversed so that any rational function can be expressed as the sum of partial fractions. Any polynomial can be rewritten as a product of linear and irreducible quadratic factors. So q(x) can be decomposed. Linear fractions have only a constant in the numerator, regardless of the number of repetitions. Quadratic fractions have linear and constant terms only.

Why useful? U S B T R I G S U B

Example 1

Example 2

Example 3

Example 4

Lecture 15 – Improper Integrals Infinite Integrals: infinity at one or both limits. Example 1

Example 2

Example 3

Example 4

Example 5 Find the volume of the solid generated by revolving the region bounded the curve and the x-axis on the interval around the x-axis. f(x) x 1 2 3 2

x 2

Lecture 16 – More Improper Integrals Discontinuous Integrands: infinite discontinuity in intervals [a, b]. Example 6

Example 7

Example 8

Example 9